Numerical analysis on seismic response and failure mechanism of articulated pile−structure system in a liquefiable site from shaking-table experiments

Abstract

This study investigates the seismic response and failure mode of a pile−structure system in a liquefiable site by employing a numerical simulation model combined with the shaking-table results of a soil−pile−structure dynamic system. The pile and soil responses obtained from the numerical simulations agreed well with the experimental results. The slopes of the dynamic shear-stress–shear-strain hysteretic curves at different positions also exhibited a decreasing trend, indicating that the shear strength of the soil in all parts of the foundation decreased. The peak acceleration of the soil and pile was not clearly amplified in the saturated sand layer but appeared to be amplified in the top part. The maximum bending moments appeared in the middle and lower parts of the pile shaft; however, the shear forces at the corresponding positions were not large. It can be observed from the deformation mode of the pile-group foundation that a typical bending failure is caused by an excessive bending moment in the middle of the pile shaft if the link between the pile top and cap is articulated, and sufficient attention should be paid to the bending failure in the middle of the pile shaft.

Graphical abstract

Keywords

numerical simulation / soil liquefaction / pile foundation / shaking-table experiment / seismic responses / failure model

1 Introduction

Pile foundations, as one of the most widely used deep foundation forms, can better adapt to complex geological conditions and various loads compared with other types of foundations and possess the advantages of a large bearing capacity, good stability, and small differential settlement; therefore, they are widely used to support bridges, high-rise buildings, and offshore structures. However, the liquefaction of saturated sand caused by earthquake loading can cause serious damage to bridges, ports, wharfs, offshore wind turbines, and other pile structures. In the 1980s, Hamada et al. [1] conducted a case study on the permanent displacement field during the Niigata earthquake and discovered that soil liquefaction can lead to large permanent ground deformation. Subsequently, the decline or even loss of pile foundation bearing capacity caused by sand liquefaction has attracted the attention of many geotechnical earthquake engineering experts and scholars.

As a column member buried deep in the ground, the dynamic response of the pile foundation under earthquake loading is not only constrained by the surrounding soil but also greatly affected by the vibration of the structure. Therefore, the seismic response of soil–pile−structure systems in liquefied sites is a very complicated problem. Combined with pile foundation damage caused by large earthquakes, such as the Wenchuan et al. [2–7], it can be seen that the soil lateral displacement [8–11], amplification effect of ground vibration in soft soil sites [12], and excessive deformation of pile foundations due to sand liquefaction [13,14] are primary reasons for the seismic failure of many damaged structures.

Some scholars believe that saturated sand liquefaction softens the soil, which results in the soil showing “non-Newtonian fluid” characteristics and leads to an evident decline in lateral constraints on pile foundations [15–17]. The stiffness of the non-liquefied soil layers is relatively high, and the lateral constraint of the flowing soil on the pile leads to shear failure and bending failure of the pile shaft in the liquefied layer or at the interface of the liquefied and non-liquefied layers [18,19]. Some studies [20,21] have shown that the interfaces between different soil layers are regarded as dangerous parts due to large bending moments and shear forces. Owing to the vibration phase difference between the structure and soil, shear failure occurs at the pile head [22,23], which is often characterized by an obvious dislocation or complete fracture of the pile top and cap. The superstructure tilts under the action of ground motion [24,25]. Some scholars hold the view that the P−Δ effect generated after tilting of the structure increases the bending moment of the pile and intensifies the overturning failure of the structure [26–30]. In general, owing to the complexity of the soil−structure dynamic interaction, the failure mechanism and failure mode of a pile foundation in a liquefiable site should be further discussed.

Experiments and numerical analysis models are the main research methods for studying the soil−structure dynamic interactions and seismic responses of pile foundations. Large shaking table and centrifuge tests have some shortcomings, such as a long test period, poor repeatability of the test conditions, and many uncertain factors in the experimental process. The numerical simulation not only provides a feasibility basis for experimental research, but also provides a powerful supplement. Therefore, numerical-simulation methods have been favored by several researchers. As a professional geotechnical engineering analysis software, FLAC (2D/3D) has powerful calculation functions and extensive simulation capabilities, particularly the ability to simulate fluid-structure interactions, which can effectively simulate the liquefaction behavior of saturated sand in liquefiable sites.

Mansour et al. [31] studied the liquefaction site and the lateral expansion flow behavior of soil caused by liquefaction site by using FLAC2D finite-difference software. Barari et al. [32] used a nonlinear stiffness degradation model to conduct fluid-solid–structure-seabed coupling analysis of a single-pile wind turbine foundation in liquefaction soil in FLAC2D finite-difference software, focusing on the seismic response mechanisms of a single pile during liquefaction in a saturated sandy soil site. The seismic responses of pore–water pressure, pile–soil relative displacement, and pile bending moments under earthquake loading with different frequencies and acceleration amplitudes were obtained and analyzed. Rayhani et al. [33,34] used the fully coupled nonlinear finite-difference program FLAC2D to simulate the mechanical characteristics of structures on uniformly layered soft soil foundations and studied the influence of soil−structure interaction on foundation motion. The results showed that the peak acceleration of soil under the structure would increase under strong soil−structure interactions. Haldar and Babu [18] simulated and discussed the soil liquefaction behaviors and soil–pile dynamic interaction using FLAC2D finite-difference program. It was found that the response of the pile structure at the liquefiable site was affected by the soil relative density, seismic frequency, seismic intensity, pile material, and pile diameter, and that the failure mode of the pile was closely related to the depth of the liquefaction layer. Comodromos et al. [35–38] studied the mechanical properties of pile groups and the influence of pile-group layout on the bearing capacity and stiffness of pile groups via FLAC3D and evaluated the influence of pile interaction on the bearing capacity and stiffness of a single pile and pile group. Hokmabadi et al. [39] conducted an experiment and used a full nonlinear three-dimensional numerical simulation model to analyze the soil−structure dynamic interaction in different sites, e.g. rigid foundation, shallow foundation, and floating pile foundation. Esfeh and Kaynia [24,40] performed a nonlinear dynamic analysis of offshore structures in saturated sand using the SANISAND constitutive model in FLAC3D. The results indicate that both monopiles and caissons in liquefiable soil deposits experience considerable rotation under earthquake shaking when liquefaction occurs. Chatterjee et al. [41,42] performed numerical simulation on a single pile in a liquefiable site or non-liquefiable soil based on the finite-difference program FLAC3D, which focused on the dynamic response of the deformation and bending moment of the pile. It was concluded that a thorough evaluation of both inertial and kinematic interactions is necessary to consider the behavior of piles at liquefied sites.

Previous studies have often focused on the macroscopic phenomenon of liquefaction, overall seismic response law, or macroscopic failure mode of pile foundations. In addition, the piles and caps were articulated for actual engineering applications. The assessment of the dynamic interaction and failure mechanism of end-bearing friction piles at a liquefaction site is not comprehensive or sufficiently clear. In addition, the articulated link between the pile and cap has rarely been discussed in previous studies on the seismic response of an entire dynamic system at a liquefiable site. Research on the dynamic interaction of pile foundation-structure systems and the failure mechanism of end-bearing friction piles in liquefaction sites needs to be further improved. In this study, combined with the results of numerical simulations and shaking-table tests, the dynamic response and failure mode of pile foundations in a liquefiable site-pile-group-structure seismic system subjected to earthquake loading were studied. First, the finite-difference software FLAC3D was used to establish a numerical simulation model of the pile−structure system at the liquefiable site. In the numerical simulation model, the articulated link between the pile top and pile cap and the actual mechanical condition of the piles were precisely described. The rationality of the numerical model was verified using the experimental results of a liquefiable soil−pile-group superstructure system. By analyzing the pore-pressure response of saturated soil, the acceleration response of the soil and structure, bending moment of the pile shaft, and seismic response of the pile−structure system in liquefiable sites were discussed. Furthermore, the variation rules of the soil shear-stress–shear-strain relationship and failure characteristics of the liquefied site were studied, and the failure mechanism and deformation mode of the pile at the liquefied site under earthquake loading were emphasized.

2 Shaking-table test design and numerical model establishment

The shaking-table experiment was conducted on a simplified structure with two lumped masses supported by a pile-group foundation at a liquefiable site [12,43]. Fig.1 shows the setup of the shaking table.

In this study, a three-dimensional finite-difference numerical analysis model of the earthquake response of a soil–pile−structure system was developed via the FLAC3D finite-difference software to simulate the shaking-table test of a soil–pile−structure system in a liquefiable site and further investigate the seismic response and failure mechanism of piles and structures after sand liquefaction.

2.1 Geometry and property

Fig.2(a) and Fig.2(b) show the site-pile–cap-structure dynamic system in the large-scale shaking-table test and the system established in the numerical simulation, respectively. As shown in Fig.2(a), in the shaking-table test, the model site consists of three layers of soil, namely, a 0.3 m clay layer, 1.2 m saturated sand layer, and 0.5 m dense sand layer, respectively, from top to bottom. The pile−structure system was composed of 2 × 2 piles, a cap, and a superstructure. The length and diameter of the pile were 1700 and 100 mm, respectively. The length embedded in the cap was 50 mm, and the length buried in the dense sand layer was 400 mm. The dimensions of the cap were 800 mm × 800 mm × 250 mm. The superstructure consisted of two lumped masses (setting at 2 and 3 m) and steel columns. A detailed description of the shaking-table tests has been introduced in Refs. [12,43]. As shown in Fig.2(a), the geometric ratio of the numerical model and model system of the shaking-table test was set to 1:1, and the pile element was used to simulate the pile-group foundation.

The geometry and discretization of the soil–pile−structure dynamic system and the positions of the measurement points are shown in Fig.3.

An elastic model was used to simulate the superstructure and cap in the numerical model, and the parameters were selected according to the experimental design and measured parameters. Before the shaking-table test, some soil was collected in the model soil container, and the density, cohesion, friction angle, permeability, and other soil parameters were measured using density tests, direct shear tests, permeability tests, and other laboratory test methods. The Mohr–Coulomb model was used to simulate both clay soil and dense sand. The soil properties of the Mohr–Coulomb constitutive model are listed in Tab.1. To ensure comparability between the numerical simulation and shaking-table test, the parameters used in this numerical simulation are the soil parameters measured from laboratory tests before the shaking-table experiment. The grain-size distribution of fine sand is shown in Fig.4.

Undoubtedly, the simulation of liquefaction occurring in the loose sand layer is the most important for the reliable prediction of the overall numerical model. To accurately describe the sand liquefaction behavior, the SANISAND constitutive model proposed by Dafalias et al. [45–47] as a relatively mature constitutive model was used to simulate saturated sand in this numerical simulation. SANISAND has been chosen in a large number of previous studies performed by many scholars to discuss the liquefaction behavior of saturated sand soils and the dynamic responses of the structural system located in liquefiable sites [45,47,48]. An advantage of this constitutive model is that a set of stress parameters can be applied to significantly different stresses and densities. In addition, some material parameters are considered the default values for most sand bodies, while other sand bodies do not need to be determined separately. According to some previous parameter calibrations for different types of soil [40,46,48], the soil material parameters of saturated sand determined in this simulation are shown in Tab.2.

In the shaking-table experiment, the E–W component of the Wolong ground motion record of the Wenchuan earthquake was selected as the input motion. The time history and Fourier spectrum of the Wolong ground motion record used in the shaking-table experiment are shown in Fig.5. The frequency components of input seismic waves and soil wave velocity would affect the numerical accuracy of wave propagation in soil; it has been shown that the mesh size should be less than 1/8–1/10 of the wavelength corresponding to the highest frequency recorded by the input seismic to accurately describe the wave propagation in the soil materials [49]. The seismic records used in this simulation are the filtered time history of the Wolong Station seismic record of the Wenchuan earthquake, and the filtering frequency is 0.1–10 Hz. The maximum unit size of the simulated middle-earth element in the vertical direction is 0.1 m, and the maximum unit size in the longitudinal and transverse directions is 0.2 m. It can be calculated that the maximum element size in the numerical model is smaller than the maximum element size required for numerical simulation accuracy.

2.2 Simulation of pile group

Because there is no specific requirement between the modeling of the structural element and the position of the solid element in FLAC3D, the pile element was used to simulate the pile group in the shaking-table test. The nodes in the pile element can establish links with the nodes of the surrounding solid grid, and the interaction between the pile and soil can be realized through this type of connection. To precisely simulate the mechanical characteristics and dynamic response of the pile in its actual state, the following two connection modes need to be paid attention to: 1) the end-bearing connection between pile tip and soil in the dense sand layer; 2) an articulated link connects the pile and the pile cap.

The piles in the shaking-table test were end-bearing friction piles. In other words, the piles are suspended in the soil deposits, and the pile tips are not connected to the bottom rigid body, while the pile tips can still bear the end-bearing force from the dense sand layer. The vertical bearing capacity of the pile is mainly determined by two factors: the friction force on the pile surface and the vertical end-bearing force of the pile tip from the soil bed, as shown in Fig.6. In the numerical simulation model, the friction action between the pile and soil was reflected by setting the friction angles of the contact surface and the other parameters of the coupling springs. The pile–soil link at the bottom of the pile element cannot reflect the vertical end bearing of the soil at the pile end. To accurately simulate the vertical end bearing of the soil at the pile tip, the pile–soil connection mode and its mechanical properties at the pile tip should be reset.

A single-pile numerical analysis model was established to determine the pile element parameters and mechanical property settings. Because the piles do not contact the clay layer, the site consists only of a saturated sand layer and a dense sand layer. The single-pile analysis model is illustrated in Fig.6. Saturated sand and dense sand were simulated using the Mohr–Coulomb constitutive model, and the joint damping scheme in FLAC3D was selected. All physical and geometric parameters, such as density, pile diameter, and site size, are in agreement with the shaking-table experiment, and the material properties of the soil are in accordance with the parameters listed in Tab.1. The normal and tangential coupling stiffnesses were calculated using Eqs. (1) and (2), respectively. The values of the normal and tangential cohesion were set to 0, and the values of the normal and tangential coupling friction angles were set to 35°, which are the same as those of the surrounding soil. The original link at the pile tip node was deleted in the model, and a new link was added with a normal yield spring in the vertical direction (Z-direction in the global coordinate system). The spring stiffness was set to 2 × 10⁵ kPa, and the yield strength was set to 3 kPa. The rotational degrees of freedom in the three directions are set as free, and the stiffness of the two translational degrees of freedom was selected according to Eq. (1) (X and Y directions in the global coordinate system). When a vertical load is applied to the pile, the maximum bearing capacity between the soil and pile is 4.3 kN by the numerical calculation model, and the total bearing capacity is 6.8 kN when considering end bearing, while the theoretical calculation results are 4.25 and 6.9 kN, respectively. Therefore, it can be shown that the selected pile element parameters in the numerical model of the single-pile bearing capacity analysis are reasonable and reliable and can be used in the numerical simulation model of the pile group.

(1)k_{\text{n}} =\frac{4G{r}_0}{1-\mu } ,

(2)k_{\text{s}} =\frac{32\left(1-\mu \right)G{r}_0}{7-8\mu },

where k_n denotes the normal coupled spring stiffness, k_s denotes the tangential coupled spring stiffness, μ denotes the Poisson ratio, G denotes the shear modulus of the soil, and r₀ denotes the diameter of the pile.

2.3 Connection between cap, pile, and interface parameters

To correctly describe the contact behavior between the pile cap and surrounding soil, it is necessary to establish a reasonable contact surface between the pile cap and soil. The normal and tangential stiffnesses of the interface were set at 2 × 105 kPa. The friction angle was 0.6 times that of the surrounding clay soil, and cohesion was 1 kPa.

In the design scheme of the shaking-table experiment, the pile cap and piles were not cast in one piece, so the connection between the cap and pile is not rigid. As shown in Fig.7(a), the pile cap has four reserved holes at a certain depth at the pile installation position. According to the test model design, the diameter of each reserved hole was 102 mm, and the diameter of each pile was 100 mm. The piles were embedded in the pile cap, so the connection between the pile and pile cap was closer to the articulated connection. The default link between the pile element and cap cannot reflect the real connection, so it is deleted first, and then a new link is set, as shown in Fig.7(b). The new link restricted translational motion in three directions, and three rotational degrees of freedom were set as free to reflect the articulated connection of the pile and pile cap.

2.4 Boundary and damping

In the shaking-table test and numerical simulation, the soil boundary should be set to minimize wave reflection at the boundary. In the shaking-table test, a unidirectional laminar shear box was selected as the soil container. In fact, a shaking-table experiment in the free field was conducted to verify whether the “model soil box boundary effect” was obvious for the model soil box used [50]. The test results demonstrate that the adopted model box effectively eliminates wave reflection at the boundary.

The influence of wave reflection on the boundary should be considered if dynamic analysis is performed. In this simulation model, the free-field boundary provided by FLAC3D was adopted, which can better eliminate the reflection of waves at the boundary of the soil, and this provided approximately the same boundary effect as that of a real site. A schematic of the free-field boundary principle is shown in Fig.8. The setting of the free-field boundary and the introduction of its working mechanism can be found in the FLAC3D User Manual [49].

The internal friction of the soil material and possible sliding at the interface will lead to damping, which should be introduced into the numerical calculation to correctly simulate the energy dissipation of the soil. FLAC3D provides various damping schemes for static and dynamic calculations. Considering the computational efficiency of the numerical model, the damping of the clay layer, dense sand layer, and structure were selected as local damping; the critical damping ratio of the clay layer and dense sand layer was 10%, and the critical damping ratio of the pile, cap, and superstructure was 3%. Rayleigh damping is employed on the saturated sand layer, and the two parameters of Rayleigh damping set in FLAC3D are 0.05 and 5, respectively.

3 Results and discussion

3.1 Excess pore-pressure response of saturated soil

Pore-pressure transducers were installed to monitor and record the entire process of the accumulation and dissipation of excess pore pressure at different depths of the saturated sand layer, including the soil far away from the pile foundation and the area adjacent to the piles in the shaking-table test. The pore-pressure transducers are denoted as W, and some data from several measuring points are used in the following analysis. The data of measuring points W4–W6, which are far away from piles and structures, and measuring points W12–W16, which are located in the center of the site and directly below the pile cap, were selected for comparative analysis, with the results at corresponding positions in the numerical model, as shown in Fig.9 and Fig.10. To clarify the sign of the soil liquefaction stated in this paper, it is stipulated that the pore-pressure ratio of saturated sand reaches 0.6 as the mark of liquefaction.

It can be seen from Fig.9 and Fig.10 that, with the increase in the input acceleration amplitude, the pore-pressure and time history curves at each measuring point accumulated rapidly, reaching peaks at approximately 6 s. The excess pore-pressure ratio in the upper soil of the saturated sand was close to 1, indicating that the soil near this area was completely liquefied. The ratios of excess pore-water pressure at the measuring points W5 and W6 were about 0.6 and 0.4, respectively; that is, pore pressures at the measuring points in the middle and lower parts of liquefiable sand were at a high level, but the soil was not completely liquefied.

The general trend of the pore–water pressures monitored in the numerical simulation model showed that the variation trends of the time history curves of excess pore-water pressure were consistent with the recorded pore pressures in the experiment, and the timings of the pore-pressure ratio peak amplitudes also roughly matched. This indicates that the experimental and numerical solutions were in good agreement and that the numerical simulation can reasonably simulate the variation rules of the pore-water pressure in the saturated sand layer.

3.2 Acceleration response of soils

The miniature acceleration sensors embedded in the Shape Acceleration Array (SAA) can be used to measure the acceleration time history of the soil and piles. Fig.11 shows the time history curves of soil acceleration at different depths from the shaking-table test and numeric simulation model under the 0.3g Wenchuan Earthquake Wolong record. The acceleration results from the shaking-table test and the numerical simulation model were compared. This shows that the acceleration waveforms and amplitudes are similar before the saturated sand is liquefied. However, the acceleration results obtained by the numerical simulation were smaller than those obtained experimentally after soil liquefaction. This is because the constitutive model of liquefied sand used in this simulation has the shortcoming of excessive damping when the strain is high. This causes the soil damping after the liquefaction of sand to be too large, which affects the propagation of the added velocity in the saturated sand layer and causes it to decrease. This is because the liquefaction-sand constitutive model used in this simulation has the disadvantage of excessive damping when the strain is considerable. The large strain after sand liquefaction results in excessive soil damping, which affects the propagation of seismic waves in the saturated sand layer and leads to a reduced acceleration response.

Fig.12 shows the comparative results between the experimental solutions of the soil-acceleration amplification coefficient measured at each measuring point of SAA2 and the numerical solutions, where the acceleration amplification coefficient is the ratio of the acceleration peaks at each measuring point to the acceleration peak of the input earthquake record.

Fig.12 indicates that the acceleration amplification coefficients of the measuring points are less than 1, which indicates that the acceleration response was not amplified at the model site. In addition, with a decrease in depth, the acceleration amplification coefficients of all measuring points first decreased and then increased. In the saturated sand layer in the middle, the peak acceleration decreases significantly, the minimum value is about 0.168g, and the attenuation is about 56% of the input ground motion. This is because the saturated sand layer in the middle of the liquefaction site liquefies under the excitation of 0.3g Wolong ground motion, which results in the softening of the soil and significant dissipation of seismic energy in the soil.

3.3 Dynamic shear stress–shear strain of soils

The dynamic shear-stress–shear-strain hysteretic curve of the soil can be obtained by inverting and calculating from the soil-acceleration time histories. Fig.13 shows the dynamic shear-stress–shear-strain hysteretic curves of the soil at different depths in the shaking-table test and numerical model. The equivalent shear modules are the slopes of the lines at the apexes of the stress-strain curves. Owing to the influence of experimental noise and errors caused by data processing, the results obtained from the experimental data were not as regular as predicted, and the hysteresis curves obtained by inversion and calculation from the experimental data were different from those obtained by numerical simulation to some extent. However, from the overall rules and shape of the stress–strain hysteretic curves, the general trend of the numerical simulation of different depths in the soil shear-stress–shear-strain hysteresis agreed with the experimental results. From a qualitative point of view, on the basis of reciprocal verification of results from the numerical model and shaking-table test, it can be seen that the numerical simulation model could effectively reflect the relatively reliable dynamic shear-stress–shear-strain relationship and the variation rule of the equivalent shear modules of soil under seismic loading.

Fig.14 illustrates the shear-stress–shear-strain curves of the soil at different depths at the liquefaction site. It can be seen that within 0–2 s, the shear-stress–shear-strain curves of soil at different depths can be thought of as straight lines in fact, indicating that the soil is in the elastic or weak nonlinear response stage, and has not yet shown strong nonlinear behavior. With the input of seismic loading, the shear stress and shear strain within 2–6 s increased rapidly, and the shear strain was about 50 times that within 0–2 s, and the shear stresses also increased nearly eight times the original stress. Within 6–10 s, the input acceleration decreased, pore–water moved upward, and the shear stress and shear strain also decreased.

Fig.15 demonstrates the variation rule of the equivalent shear modules of soils at various depths near and far piles in the liquefaction site within 0–10 s. It was shown that the shear modulus of soil at different depths first decreased and then increased, while the soil stiffness gradually increased from shallow soil to deep soil. When the soil shear modulus within 0–6 s was lower than the initial value, the shear strength of the saturated sandy soil after liquefaction decreased significantly. Within 4–6 s, that is, the period near the timing of peak acceleration, the shear modules of the soil at all depths were reduced to the minimum. After this period, the pore-water pressure remained at a very high level, but the soil regained some of its stiffness. In the process of seismic loading, the shear modules of soil near piles were slightly different at various depths, which indicated that the pile–soil dynamic interaction also has a certain influence on the shear modules of soil in the vicinity of piles.

3.4 Acceleration response of piles and the superstructure

Fig.16 shows the numerical simulation and test acceleration time history curves of the measurement points of the liquefied non-free-field pile foundation. The measured curves from SAA1-0 to SAA1-5 are the acceleration time histories of the pile body, and the curve recorded by SAA1-6 is the acceleration time history of the cap. Fig.17 shows the amplification coefficient distribution of the pile acceleration. By comparing the results shown in Fig.16 and Fig.17, it can be seen that the pile acceleration results obtained in the test are consistent with the numerical simulation results. In fact, it was demonstrated that the settings of the pile elements and parameters in the numerical analysis model were reasonable and reliable. It can be seen from Fig.14 that the peak acceleration of the saturated sand layer attenuates to 0.189g, which is approximately 63% of the input ground motion. The acceleration amplification coefficient showed a trend of an obvious decrease first and then amplification from bottom to top, which was consistent with the rule of soil acceleration.

Fig.18 illustrates the acceleration time history curves of the superstructure base and two concentrated masses in the experimental and numerical analysis models. According to the overall rules shown in Fig.18, the acceleration amplitude of the superstructure (measuring point X1a) in the soil–structure interaction system was smaller than that of the input ground motion because the liquefied soil did not significantly amplify the acceleration response. The predominant frequency of the Wolong ground motion used in the shaking-table test was 4.69 Hz, which was significantly higher than the first-order vibration frequency of the superstructure of 1.33 Hz. In addition, part of the energy of the superstructure dissipated in the liquefied soil, so the acceleration response of the two concentrated masses of the superstructure (corresponding to measurement points X2a and X3a) were all smaller.

3.5 Bending-moment response of pile shaft and failure mode

In the shaking-table experiment, strain gauges were installed on two steel bars, as shown in Fig.19, and the bending moments of the pile shaft were calculated using the corresponding tensile and compressive strains. The moment time history curves at nine measuring points of the pile shaft from the shaking-table test and the numerical simulation model are shown in Fig.20. A comparison of the experimental and numerical simulation results showed that the overall trend of the bending moments from the numerical simulation was basically consistent with the experimental results. After the saturated sand was liquefied, the bending moments of the pile in the numerical simulation model were larger than those in the experiment, probably because the setting stiffness of the interface between the piles and soil could not reflect the influence of the soil modulus change after sand liquefaction.

Fig.21(a) shows the distribution of the maximum and minimum bending moments of the pile shaft in the numerical simulation and experiment and the residual bending moments along the soil depth at the end of the seismic loading (at the timing of 50 s). From the results shown in the figure, it can be observed that the bending moment envelope diagram from the numerical simulation had the same trend as the experimental results; that is, the numerical simulation results reasonably reflected the bending moment distribution of the piles in the shaking-table test. The maximum bending moment of the pile shaft first increased and then decreased along the soil depth from bottom to top. In addition, the bending moment near the pile top was very small in both the experiment and numerical simulation, which indicated that the piles and pile cap were articulated, and the connection mode setting between the piles and pile cap in the numerical model effectively reflected the real response in the shaking-table test.

Fig.21(b) shows the distribution of the shear forces of the pile shaft at different positions in the numerical simulation and shaking-table experiment. The shear forces were large near the interface between different soil layers but small at the position where the pile bending moments were the largest. In addition, the shear force at the junction of the piles and pile cap was very small. The overall rules of shear force also showed that the interaction caused by soil movement and deformation at the liquefied site had a predominant influence on the shear force responses of piles rather than the inertia effect caused by the superstructure.

Fig.22(a) and Fig.22(b) show schematic diagrams of the typical deformation modes of piles obtained from the numerical calculation model and shaking-table experiments [12,43]. It can be observed that there was an obvious bending deformation in the middle of the piles. Saturated sand is liquefied under seismic loading, and the stiffness of the upper soil is greatly reduced, which decreases the lateral constraint of the soil to the pile shaft. Subsequently, structural vibration and soil lateral displacement caused the deformation and instability of piles after the soil liquefied under earthquake loading. This agrees with the results obtained from the shaking-table experiment [44]. Combined with the pile bending moment and shear force distribution shown in Fig.21(a) and Fig.21(b), it can be seen that the pile deformation in Fig.22 was a typical bending deformation. If the piles are damaged, it should be a typical bending failure caused by an excessive bending moment in the middle of the pile shaft. This also shows that when the link between the pile top and pile cap is articulated, sufficient attention should be paid to the bending failure in the middle of the pile.

4 Conclusions

Combined with a large-scale shaking-table experiment of a soil–pile–group-superstructure system, a numerical simulation model for the seismic response of a pile-group-superstructure system in a liquefiable site was established using the FLAC3D finite-difference software. Strong nonlinear behavior and large deformation of soil after sand liquefaction were considered in this numerical model, and the articulated connection between piles and the pile cap and the function of friction and end-bearing capacity of piles were precisely simulated. The validity of the numerical model was verified by comparing the seismic response and failure mode of the soil–pile-structure system. In addition, the rules for the development of the pore-water pressure, acceleration response, soil dynamic shear-stress-shear-strain relationship, and bending moments of the pile shaft are discussed in detail. Site failures after soil liquefaction and pile bending deformation modes were highlighted in this study. The following are some important conclusions.

1) Overall, the numerical simulation results were in good agreement with the experimental results. The refined numerical analysis model established in this study is an effective method for analyzing the strong nonlinear behavior of sites after saturated sand liquefaction under earthquake loading and the influence of site failure on structural failure modes.

2) Under the seismic loading of the 0.3g Wolong earthquake record, the pore-pressure ratio increased rapidly and the soil liquefied. The acceleration response of the soil at each measuring point showed that the site did not amplify in the liquefied soil layer, and the acceleration amplification coefficient first decreased and then subsequently increased from the bottom to the soil surface.

3) After the saturated sand liquefied, the slopes of the shear-stress–shear-strain relationship of the soil decreased; that is, the equivalent shear modulus of the soil decreased, and the soil softened.

4) The acceleration response of the pile showed no obvious amplification from the pile tip to the top, which is similar to the soil-acceleration rule. In addition, the acceleration response of the two concentrated masses of the superstructure was significantly smaller than that at the base of the superstructure, which indicated that the structural energy dissipated significantly in the soils and that the soil–pile–structure dynamic interaction reduced the acceleration response of the superstructure.

5) The pile bending moments were extremely small at the pile top and tip, and the maximum pile bending moment appeared near the middle of the pile. The shear forces were large at the interfaces between the different soil layers but not in the middle of the pile shaft.

6) The deformation models of the pile group deduced from the numerical simulation model and experimental results also showed that obvious bending deformation occurred in the pile shaft, indicating that piles used in a liquefied site would lose part of the lateral bearing capacity of the soil and cause typical bending failure, combined with the discussion of the dynamic responses of the soil above.

5 Summary

In this study, seismic response and interaction analyses of an articulated pile group used in a liquefaction site were performed. However, the research in this study does not involve the establishment of an effective simplified analysis method and can only provide a limited reference for practical engineering design and guidance. This will be important content for the authors to study further in the future.

Many aspects of the dynamic response and interaction of piles will require more systematic study in the future. From the perspective of pile type, most previous studies only focused on reinforced concrete piles, and there is a lack of research on prestressed piles, H-shaped steel piles, helical piles, and large-diameter piles commonly used in ocean engineering. From the perspective of the pile formation method, current research on the seismic response and failure mode of piles is limited to cast-in piles or steel piles in uniform fields, ignoring the influence of the pile formation method on the working and failure mechanisms of piles under seismic loads. In fact, the compaction effect of prefabricated piles, effect of pile boots, and stress relaxation phenomenon of boreholes are rarely considered in research on the earthquake failure mechanisms of piles and pile–soil interactions. All of these are critical areas of investigation for seismic research on pile foundations in saturated sandy soils.

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